The understanding of co-movements, dependence, and influence between variables of interest is key in many applications. Broadly speaking such understanding can lead to better predictions and decision making in many settings. We propose Quantile Graphical Models (QGMs) to characterize prediction and conditional independence relationships within a set of random variables of interest. Although those models are of interest in a variety of applications, we draw our motivation and contribute to the financial risk management literature. Importantly, the proposed framework is intended to be applied to non-Gaussian settings, which are ubiquitous in many real applications, and to handle a large number of variables and conditioning events.
We propose two distinct QGMs. First, Condition Independence Quantile Graphical Models (CIQGMs) characterize conditional independence at each quantile index revealing the distributional dependence structure. Second, Prediction Quantile Graphical Models (PQGMs) characterize the best linear predictor under asymmetric loss functions. A key difference between those models is the (non-vanishing) misspecication between the best linear predictor and the conditional quantile functions.
We also propose estimators for those QGMs. Due to high-dimensionality, the two distinct QGMs require different estimators. The estimators are based on high-dimensional techniques including (a continuum of) L1-penalized quantile regressions (and low biased equations), which allow us to handle the potential large number of variables. We build upon a recent literature to obtain new results for valid choice of the penalty parameters, rates of convergence, and condence regions that are simultaneously valid.
We illustrate how to use QGMs to quantify tail interdependence (instead of mean dependence) between a large set of variables which is relevant in applications concerning with extreme events. We show that the associated tail risk network can be used for measuring systemic risk contributions. We also apply the framework to study international financial contagion and the impact of market downside movement on the dependence structure of assets’ returns.